Add more work on part 11 of compiler series

content/blog/11_compiler_polymorphic_data_types.md

@@ -104,3 +104,117 @@ In effect, they take zero arguments and produce types (themselves).
 
 Polytypes (type schemes) in our system can be all of the above, but may also include a "forall"
 quantifier at the front, generalizing the type (like \\(\\forall a \\; . \\; \\text{List} \\; a \\rightarrow \\text{Int}\\)).
+
+Let's start implementing all of this. Why don't we start with the change to the syntax of our language?
+We have complicated the situation quite a bit. Let's take a look at the _old_ grammar
+for data type declarations (this is going back as far as [part 2]({{< relref "02_compiler_parsing.md" >}})).
+Here, \\(L\_D\\) is the nonterminal for the things that go between the curly braces of a data type
+declaration, \\(D\\) is the nonterminal representing a single constructor definition,
+and \\(L\_U\\) is a list of zero or more uppercase variable names:
+
+{{< latex >}}
+\begin{aligned}
+L_D & \rightarrow D \; , \; L_D \\
+L_D & \rightarrow D \\
+D & \rightarrow \text{upperVar} \; L_U \\
+L_U & \rightarrow \text{upperVar} \; L_U \\
+L_U & \rightarrow \epsilon
+\end{aligned}
+{{< /latex >}}
+
+This grammar was actually too simple even for our monomorphically typed language!
+Since functions are not represented using a single uppercase variable, it wasn't possible for us
+to define constructors that accept as arguments anything other than integers and user-defined
+data types. Now, we also need to modify this grammar to allow for constructor applications (which can be nested!)
+To do so, we will define a new nonterminal, \\(Y\\), for types:
+
+{{< latex >}}
+\begin{aligned}
+Y & \rightarrow N \; ``\rightarrow" Y \\
+Y & \rightarrow N
+\end{aligned}
+{{< /latex >}}
+
+We make it right-recursive (because the \\(\\rightarrow\\) operator is right-associative). Next, we define
+a nonterminal for all types _except_ those constructed with the arrow, \\(N\\).
+
+{{< latex >}}
+\begin{aligned}
+N & \rightarrow \text{upperVar} \; L_Y \\
+N & \rightarrow \text{typeVar} \\
+N & \rightarrow ( Y )
+\end{aligned}
+{{< /latex >}}
+
+The first of the above rules allows a type to be a constructor applied to zero or more arguments
+(generated by \\(L\_Y\\)). The second rule allows a type to be a placeholder type variable. Finally,
+the third rule allows for any type (including functions, again) to occur between parentheses.
+This is so that higher-order functions, like \\((a \rightarrow b) \rightarrow a \rightarrow a \\),
+can be represented.
+
+Unfortunately, the definition of \\(L\_Y\\) is not as straightforward as we imagine. We could define
+it as just a list of \\(Y\\) nonterminals, but this would make the grammar ambigous: something
+like `List Maybe Int` could be interpreted as "`List`, applied to types `Maybe` and `Int`", and
+"`List`, applied to type `Maybe Int`". To avoid this, we define a "type list element" \\(Y'\\),
+which does not take arguments:
+
+{{< latex >}}
+\begin{aligned}
+Y' & \rightarrow \text{upperVar} \\
+Y' & \rightarrow \text{lowerVar} \\
+Y' & \rightarrow ( Y )
+\end{aligned}
+{{< /latex >}}
+
+We then make \\(L\_Y\\) a list of \\(Y'\\):
+
+{{< latex >}}
+\begin{aligned}
+L_Y & \rightarrow Y' \; L_Y \\
+L_Y & \rightarrow \epsilon
+\end{aligned}
+{{< /latex >}}
+
+Finally, we update the rules for the data type declaration, as well as for a single
+constructor:
+
+{{< latex >}}
+\begin{aligned}
+T & \rightarrow \text{data} \; \text{upperVar} \; L_T = \{ L_D \} \\
+D & \rightarrow \text{upperVar} \; L_Y \\
+\end{aligned}
+{{< /latex >}}
+
+Now that we have a grammar for all these things, we have to implement
+the corresponding data structures. We define a new family of structs,
+extending `parsed_type`, which represent types as they are
+received from the parser. These differ from regular types in that they
+do not require that the types they represent are valid; validating
+types requires two passes, which is a luxury we do not have when
+parsing. We can define them as follows:
+
+{{< codeblock "C++" "compiler/11/parsed_type.hpp" >}}
+
+We define the conversion function `to_type`, which requires
+a set of type variables quantified in the given type, and
+the environment in which to look up the arities of various
+type constructors. The implementation is as follows:
+
+{{< codeblock "C++" "compiler/11/parsed_type.cpp" >}}
+
+With this definition in hand, we can now update the grammar in our Bison file.
+First things first, we'll add the type parameters to the data type definition:
+
+{{< codelines "plaintext" "compiler/11/parser.y" 127 130 >}}
+
+Next, we add the new grammar rules we came up with:
+
+{{< codelines "plaintext" "compiler/11/parser.y" 138 163 >}}
+
+Finally, we define the types for these new rules at the top of the file:
+
+{{< codelines "plaintext" "compiler/11/parser.y" 43 44 >}}
+
+{{< todo >}}
+Nullary is not the right word.
+{{< /todo >}}